Mean and Variance for Random Variables X>0

Question

I am looking to understand mean and variance for random variables a little better. Can someone help me with some examples or guide me in the right direction?

I am interested in seeing the differences between finite and infinite so I would like help with coming up with examples of random variables X>0 with:

1. Finite mean and finite variance

2. Finite mean but infinite variance

3. Infinite mean and infinite variance.

Answer 1

Simple examples: density function $f_n(x)=\frac{c_n}{1+x^n}$ for $x\ge 0$ and $f_n(x)=0$ otherwise. For $n=2$ mean and variance are infinite, for $n=3$ finite mean and infinite variance, for $n=4$ mean and variance finite.

$c_n$ defined so that $\int_0^\infty f_n(x)dx=1$.